Spectral density of the Dirac-Ginsparg-Wilson operator, chiral $U(1)_A$ anomaly, and analyticity in the high temperature phase of $QCD$ (2304.14725v2)

Abstract: Using general properties of the $Q=0$ topological sector we previously argued that a vector-like theory, with chiral $U(1)_A$ anomaly, and exact non-Abelian chiral symmetry, should exhibit divergent susceptibilities in the chiral limit, the two-flavor Schwinger model being a paradigmatic example of the realization of this scenario. Two flavor $QCD$ at $T>T_c$ satisfies all the above conditions, and it is also expected that the $U(1)_A$ axial symmetry remains effectively broken in its high temperature phase. Therefore we would expect a non-analyticity in the quark mass dependence of the free energy density, in contrast with the Dilute Instanton Gas Approximation (DIGA) prediction. We investigate in this work whether the aforementioned results can also be reproduced making only use of standard properties of the spectral density of the Dirac operator, without having to resort to general properties of the $Q=0$ topological sector. We show that the only way to derive a non-trivial $\theta$-dependence, and an analytical free energy density in $QCD$ with two degenerate flavors is that the spectral density, $\rho\left(\lambda,m\right)$, of the absolute value of the non-zero modes of the Dirac-Ginsparg-Wilson operator develops a $m^2\delta(\lambda)$ function in the thermodynamic limit. This is the expected result in the DIGA, where interactions between instantons in the dilute gas are fully neglected. However, at temperatures close to $T_c$ the interaction between instantons should become non-negligible, and the splitting from zero of the near-zero modes, which has been neglected in the DIGA, should be taken into account. Therefore we expect that the $m^2\delta(\lambda)$ contribution to the spectral density is no longer correct at these temperatures, and that the free energy density becomes a non-analytic function of the quark mass.some clarifications added